jet set 67

F.
– University of Florence - ISMD 2003
STATISTICAL HADRONIZATION
and MICROCANONICAL ENSEMBLE
F. B., L. Ferroni, hep-ph 0307061 and a paper to appear
MOTIVATION: a Monte-Carlo code for the statistical model
OUTLINE
Introduction
Microcanonical hadron-resonance gas ensemble
Monte-Carlo sampling algorithm
Preliminary results and conclusions
F.
ISMD 2003
Clusters: four-momentum P, charges Q and volume V
Statistical hadronization: any multi-hadronic state
compatible with cluster quantum numbers is equally likely
MICROCANONICAL ENSEMBLE
F.
ISMD 2003
AIM : MONTE CARLO SAMPLING OF THE
HADRON GAS MICROCANONICAL PHASE SPACE
sample the single hadronic channel first, then assign momenta
P state
P
N 1,
,NK
4
,NK
N 1,
,NK
,NK
N 1,
states
P state
P
4
states
phase space volume or
microcanonical partition function
N 1,
statistical weight of a
single channel
Nj : number of particles for the jth hadronic species
Q , Q state
F.
ISMD 2003
General expression for a cluster with volume V and four-momentum P
as a cluster expansion
1
j
h
j 1
iP x
n
j
n
j
h
1
n
j
h
n j hn
n
j
1
j
1
!
#
1
i
1
4
P
pi
3
i
N
d pi
3
j
N
2J i 1
Ni
N
N
V
2
"
N
j
n
n
j
j
inx p
In the limit of Boltzmann statistics, with
,NK
x
j
j
inx
d p exp
0
3
3
K
N 1,
n
h
nj
1
N
3
N
zj
2
1 V
i
d xe
0
x
n
zj
2J j
with
2
0
dx
4
lim
,NK
N 1,
1
h
Nj
K
i
i
Calculable analytically only in the NR and UR regimes
Nj
hn
j
F.
ISMD 2003
The problem of calculating
N 1,
, N K for relativistic massive particles with
suitable numerical methods tackled by several authors in the '50s
None of several devised approximations satisfactory for all the allowed
channels
Cerulus and Hagedorn (Suppl. N. Cim. IX (1958) 646) succesfully implemented
a Monte-Carlo method
600
500
πππ π
Resolution=4.7%
0 0 + -
π π π π ππ
Resolution=4.8%
0 0 + + - -
ηηηη
Resolution=2.7%
with
QS
dN/dΩ
400
10
300
200
100
0
600
500
πππ π
Resolution=4.8%
0 0 + -
π π π π ππ
Resolution=5.3%
0 0 + + - -
ηηηη
Resolution=2.9%
10
Quantum statistics
Classical statistics
-1
-2
400
dN/dΩ
no
QS
N π0 channel - 2 GHz clock rate
CPU time/channel (s)
Ω distribution with 1000 MC steps
300
10
-3
200
100
0
2500
3000
3500 800
2
x 10
1000
1200
3
x 10
6000
7000
10
Ω
-4
4
6
8
10
12
14
16
18
20
N
F.
ISMD 2003
MAIN DIFFICULTY: SIZE
Number of light-flavoured hadronic species (PDG 2002):
271
10 10
Nmax = M / <m>
max
271
N max !
Time needed to calculate all channels
diverges exponentially with M
10 8
Number of channels
N channels
271
N
If Nmax
10 9
10 7
10 6
10 5
10 4
10 3
10 2
1
2
3
4
5
M (GeV)
Need a non-deterministic sampling algorithm
F.
ISMD 2003
METROPOLIS ALGORITHM
First proposed by Werner and Aichelin: Phys. Rev. C 52 (1995) 1584
Random walk in the channel
(or multihadronic configuration) space
m
Pm t w m
Pn t w n
Pm t
1
Pm t
Master equation
n
m
at equilibrium
if
n
Pm
w n m
w m n
n
m
and a channel can be extracted
Design goal: minimize the number of steps needed to reach equilibrium
F.
ISMD 2003
Start from fast-sampling distribution as close as possible to the final one
Clever choice of acceptance and proposal matrix so as to maximize the transition
probability
m
A n
m
N
exp
m
j
nj
nj
N j!
1
j
T n
<nj> = canonical averages
T n
n
m
set T as the grand-canonical limit
of the multihadron multiplicity
distribution, the multi-Poisson
m
min 1,
n
T m
m
T n
m
OPTIMAL CHOICE for T
A n
OPTIMAL CHOICE
for A once T is known
m
m A n
T n
m
w n
m
FAST CONVERGENCE TO EQUILIBRIUM !
P2
0.7
0.65
0.6
0
2000
4000
x 10
Step
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0.055
-2
0.05
0.04
-3
0.035
0.03
0
-2
2000
4000
Step
P5
0.06
P4
___ Microcanonical
----- Poisson
0.55
0.045
10
0.44
0.42
0.4
0.38
0.36
0.34
0.32
0.3
0.28
0.26
0.75
P(N)
10
0.8
P3
M=2 GeV, free charges, M/V=0.4 GeV/fm
-1
ISMD 2003
'simple' updating rule
3
10
F.
0
2000
4000
0
2000
4000
Step
poissonian updating rule
-4
0.68
P3
0.35
P2
10
Step
0.34
0.66
0.33
10
-5
0.64
0.32
0.62
0.31
0.3
0.6
2
4
6
8
10
12
20
30
40
50
0.28
0
10
20
30
Step
14
N
Total multiplicity distribution
10
0.07
0.065
0.06
0.055
0.05
0.045
0.04
0.035
0.03
40
50
Step
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0
P5
0
0
P4
10
0.58
-6
0.29
0
10
20
30
40
50
Step
0
10
20
30
40
50
Step
F.
ISMD 2003
Further advantages of the poissonian updating rule
Efficient sampling
Convergence faster for larger clusters (approach grand-canonical limit)
Low auto-correlation lenght
→
Q=0
M=2 GeV
1.06
CPU time/event (s)
A(h)
1.08
1.04
Resonance broadening
1.02
Quantum statistics
1
Free charges
Boltzmann statistics
10
1
A(h)
0.98
1.08
Free charges
1.06
1.04
No resonance broadening
1.02
Boltzmann statistics
1
10
10
-1
-2
1
2
3
4
5
6
7
8
9
10 11 12 13
M (GeV)
0.98
0
20
40
60
80
100
120
140
h
F.
ISMD 2003
COMPARISON BETWEEN MICRO AND CANONICAL AVERAGES
Statistical model calculations so far have been done within the canonical
ensemble assuming the equivalence between the set of clusters and one
global cluster
TEMPERATURE introduced through a saddle-point approximation
exp M T Z can Q
P
Q , Q state
P state
P
4
for large M and V
M ,0
2
3
m T
2
j
p
2
d p exp
3
HOW LARGE M AND V SHOULD BE ?
Z can Q
1 V
2J j
canonical average
multiplicity
nj
states
Z can Q
qj
micro
n
can
can
30
4GeV
8GeV
12GeV
25
ISMD 2003
with Q = 0 and M/V= 0.4 GeV/fm3
∆%
∆%
n
n
Study of
F.
20
0
20
15
-20
10
-40
5
-60
4GeV
0
8GeV
12GeV
-80
-5
φ
p n Λ Σ Σ Σ ∆ ∆ ∆ ∆ Ξ Ξ Σ Σ Σ Ξ Ξ Ω
!
+
η’
*0
K
*±
K
ω
±
ρ
0
ρ
see also F. Liu et al.
hep-ph 0304174
η
0
K
±
K
±
π
0
π
0
-
++ +
0
-
0
-
*+ *0 *-
*0 *-
-
n
micro
n
n
can
ISMD 2003
with Q = 0 and M/V= 0.4 GeV/fm3
can
∆%
Study of
F.
Increasing the number of
degrees of freedom gets
microcanonical and canonical
averages closer
35
30
π
25
All hadrons
20
Only π
0
0
15
10
5
0
-5
2
4
6
8
10
12
M (GeV)
F.
ISMD 2003
10
Free charges,
10
M/V=0.4 GeV/fm3
10
10
P(N)
P(N)
MICROCANONICAL VS CANONICAL MULTIPLICITY DISTRIBUTIONS
-1
〈N〉=2.39
D=0.57
-2
M=2 GeV
10
10
-3
10
-4
0
5
10
10
15
〈N〉=4.30
D=0.95
M=4 GeV
-1
-2
-3
-4
0
5
10
15
10
10
10
10
〈N〉=8.33
D=1.32
M=8 GeV
-1
N
P(N)
P(N)
N
1
10
〈N〉=12.38
D=1.73
M=12 GeV
-1
-2
10
-3
-4
0
5
10
15
20
N
10
-2
-3
5
10
15
20
25
N
F.
ISMD 2003
Preliminary simplified version of an e+e- event generator
Run parton shower program
Assign final (virtual) partons a volume V = M/0.35 GeV/fm3 (and γS = 0.65)
P(N)
Hadronize the clusters with Metropolis algorithm
Check overall conservation of charge (accept or reject)
10
-1
OPAL
Model
Charged track multiplicity
distribution at
√s = 91.2 GeV
10K events with
parton shower in Jetset7.4
10
10
-2
-3
0
10
20
30
40
50
60
N
%$
!(
?
=
;
=
=
<=
=
;
=
%
9
:
>
=
=
<=
!$
D
?
=?
=
=
=
>
=
=
=
9
9
<=
:
=
=
:
'
?
9
C
A
=
=
8
=
=
=
<=
A
=
=
=
C
A
=
=
<=
<=
9
9
9
9
9
?
D
R
>=
=
=
<=
>@
A
=
=
>;
=
=
<=
C
A
=
=
C
=
<=
C
;
=
=
8
AQ
C
9
1
@
N
7
C
;
=
C
9
9
7
=>
;
=
=
R
Q;
C
>
C
SL
&
9
9
9
9
9
9
O
#
9
9
9
9
9
9
O
?
;
;
=
=
<=
C
>
>
=
=
;
;
=
=
<=
C
>
>
=
=
;
;
=
<=
:
@
=
=
KE
"
9
9
9
9
9
9
R
?
!
@
@
=
=
<=
A
A
=
;
=
>
@
=
=
<=
:
=
=
;
=
>=
=
<=
A
;A
=
=
Q;
>
O N
P
9
9
9
9
9
9
@
=
=
<=
D
?
?
;D
;
;
=
@
@
=
=
<=
;
A
=
;
=
C
A
=
=
<=
8
=
=
M
9
9
9
9
9
9
?
D
@=
=
=
<=
8
=
A
=
=
8
>
=
=
<=
:
:
A
=
=
=
<=
:
:
;
=
H L
9
9
9
9
9
9
;D
=
=
<=
>
D
D
D
?
;D
@
=
>
=
=
<=
C
@
=
>
;
=
<=
=
@
=
KE
B
9
9
9
9
9
9
;D
=
=
<=
D
D
?
?
AD
@
=
>
=
=
<=
=
C
@
=
>
>
=
<=
C
@
=
KE
7
9
9
9
9
9
9
J
D
8
A
=
=
<=
:
:
8
=
8
=
=
<=
:
D
?
AD
8
=
8
C
=
<=
>
=
;
9
9
9
9
9
9
I
;?
=
=
<=
C@
=
=
;
;?
=
=
<=
:@
8
8
=
>
>
<=
=>
;
B
9
9
9
9
9
9
I
=
;
=
<=
=C
=
;
=
;
=
<=
>
C
=
;
@
;
<=
=
;
D
7
9
9
9
9
9
9
H
8
=
=
<=
C
A
A
A
=
8
=
=
<=
A
A
A
=
:
A
=
<=
8
C
=
;
D
D
9
9
9
9
9
9
=
;
=
<=
C
@
=
;
=
;
=
9
9
D
?
?
@?
=
=
<=
>;
:
=
;
=
=
C
<=
8
=
;
:
>
=
<=
9
9
9
<=
>
;
;
C
?
9
:
>
=
;
:
8
=
=
<=
:
@
=
;
E
FG7
E
B
9
9
9
9
9
9
6
=>
=
<=
>;
:
A
>;
=
<=
:
A
=
;
<=
C
=
A
D
D
B
9
;
@
=
<=
@
A
;
@
=
<=
?
@A?
8
8
>
9 8
9
9
9
<=
:;
9 8
6
7
/0
.
1
23
43
5
43
+ )
.
-,
*+)
F.
ISMD 2003
Average multiplicities
GSPS: Parton shower developed by
R. Ugoccioni, A. Giovannini, S. Lupia
LPS: Parton shower
in JETSET 7.4
C
F.
ISMD 2003
Conclusions and Outlook
Towards a Monte-Carlo code for the statistical model
of hadronization. Next steps: inclusion of particle momenta
and BEC (L. Ferroni, T. Gabbriellini, A. Keranen, collaboration with Nantes)
Easily matcheable to parton shower cascade programs with
pre-confinement cluster formation (HERWIG)
Comparison between micro and canonical ensemble:
for M >~ 8 GeV average multiplicities sufficiently close
Many more tests possible